Příklad 2.2

Spočtěte \(\large\lim_{x\to\infty}\frac{x}{\ln^2(x^2)}\)

$$\Large \begin{aligned}

& \lim_{x\to\infty}\frac{x}{\ln^2(x^2)} \

=& \frac{\lim_{x\to\infty}x}{\lim_{x\to\infty}\ln^2(x^2)} \

=& \frac{\infty}{\infty}

\\ &\text{It's L'Hospital Time:} \\

&= \lim_{x\to\infty}\frac{1}{\frac{8}{x}\ln(x)} \

&= \lim_{x\to\infty}\frac{x}{8\ln(x)} \

&= \frac{1}{8}\lim_{x\to\infty}\frac{x}{\ln(x)} \

&= \frac{1}{8}\lim_{x\to\infty}\frac{\infty}{\infty} \

\\ &\text{It's L'Hospital Time:} \\

&= \frac{1}{8}\lim_{x\to\infty}\frac{1}{\frac{1}{x}} \

&= \frac{1}{8}\lim_{x\to\infty}x \

&= \frac{1}{8}\infty \

&= \boxed{\infty} \end{aligned} $$